A multigrid method for linear systems stemming from the Galerkin B-spline discretization of classical second-order elliptic problems is considered. The spectral features of the involved stiffness matrices, as the fineness parameter h tends to zero, have been deeply studied in previous works, with particular attention to the dependencies of the spectrum on the degree p of the B-splines used in the discretization process. Here, by exploiting this information in connection with tau-matrices, we describe a multigrid strategy and we prove that the corresponding two-grid iterations have a convergence rate independent of h for p=1,2,3. For larger p, the proof may be obtained through algebraic manipulations. Unfortunately, as confirmed by the numerical experiments, the dependence on p is bad and hence other techniques have to be considered for large p.
Donatelli, M., Garoni, C., Manni, C., Serra-Capizzano, S., Speleers, H. (2015). Two-grid optimality for Galerkin linear systems based on B-splines. COMPUTING AND VISUALIZATION IN SCIENCE, 17(3), 119-133 [10.1007/s00791-015-0253-z].
Two-grid optimality for Galerkin linear systems based on B-splines
Garoni C.;Manni C.;Speleers H.
2015-06-01
Abstract
A multigrid method for linear systems stemming from the Galerkin B-spline discretization of classical second-order elliptic problems is considered. The spectral features of the involved stiffness matrices, as the fineness parameter h tends to zero, have been deeply studied in previous works, with particular attention to the dependencies of the spectrum on the degree p of the B-splines used in the discretization process. Here, by exploiting this information in connection with tau-matrices, we describe a multigrid strategy and we prove that the corresponding two-grid iterations have a convergence rate independent of h for p=1,2,3. For larger p, the proof may be obtained through algebraic manipulations. Unfortunately, as confirmed by the numerical experiments, the dependence on p is bad and hence other techniques have to be considered for large p.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.